Efficient algorithms for the evaluation of the eigenvalues of (block) banded Toeplitz matrices

Abstract

Let A be an n × n n \times n banded block Toeplitz matrix of bandwidth k with m × m m \times m blocks having entries in a field F. We present algorithms for computing p ( λ ) = det ( A − λ I ) p(\lambda ) = \det (A - \lambda I) as well as the ratio p ( λ ) / p ′ ( λ ) p(\lambda )/p’(\lambda ) , where p ′ ( λ ) p’(\lambda ) is the first derivative of p ( λ ) p(\lambda ) with respect to λ \lambda , in roughly ( 3 / 2 ) k 2 log ⁡ n + O ( k 3 ) (3/2){k^2}\log n + O({k^3}) block multiplications. If the field F supports FFT, then the cost is reduced to O ( ( m 2 k log ⁡ k + m 3 k ) log ⁡ n + k 3 m 3 ) O(({m^2}k\log k + {m^3}k)\log n + {k^3}{m^3}) scalar multiplications. The algorithms generalize an algorithm given by W. Trench for computing p ( λ ) p(\lambda ) in the case m = 1 m = 1 in roughly k log ⁡ n + O ( k 3 ) k\log n + O({k^3}) multiplications and rely on powering a companion matrix associated with the linear recurrence relation representing the original problem.